# UMAP Is Spectral Clustering on the Fuzzy Nearest-Neighbor Graph

arXiv: https://arxiv.org/abs/2602.11662 · HTML: https://mlsanity.com/paper/2602.11662/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 24 references that are in the map.
- First posted on arXiv 12 Feb 2026.

## Builds on (most cited first, 10 of 10)

- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Provable Guarantees for Self-Supervised Deep Learning with Spectral Contrastive Loss](https://mlsanity.com/paper/2106.04156/) (2021)
- [Contrastive and Non-Contrastive Self-Supervised Learning Recover Global and Local Spectral Embedding Methods](https://mlsanity.com/paper/2205.11508/) (2022)
- [On UMAP's true loss function](https://mlsanity.com/paper/2103.14608/) (2021)
- [Your Contrastive Learning Is Secretly Doing Stochastic Neighbor Embedding](https://mlsanity.com/paper/2205.14814/) (2022)
- [A Probabilistic Graph Coupling View of Dimension Reduction](https://mlsanity.com/paper/2201.13053/) (2022)
- [Interpretable Dimensionality Reduction by Feature Preserving Manifold Approximation and Projection](https://mlsanity.com/paper/2211.09321/) (2022)

## Abstract (arXiv)

UMAP (Uniform Manifold Approximation and Projection) is among the most widely used algorithms for non linear dimensionality reduction and data visualisation. Despite its popularity, and despite being presented through the lens of algebraic topology, the exact relationship between UMAP and classical spectral methods has remained informal. In this work, we prove that UMAP performs spectral clustering on the fuzzy k nearest neighbour graph. Our proof proceeds in three steps: (1) we show that UMAP's stochastic optimisation with negative sampling is a contrastive learning objective on the similarity graph; (2) we invoke the result of HaoChen et al. [8], establishing that contrastive learning on a similarity graph is equivalent to spectral clustering; and (3) we verify that UMAP's spectral initialisation computes the exact linear solution to this spectral problem. The equivalence is exact for Gaussian kernels, and holds as a first order approximation for UMAP's default Cauchy type kernel. Our result unifies UMAP, contrastive learning, and spectral clustering under a single framework, and provides theoretical grounding for several empirical observations about UMAP's behaviour.